16 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA
Let (Ea)ael be any family of uniformizable spaces, and let E = fj Ea.
ael
For each ael, let (dat^)^eL<x ^e a family of pseudo-distances defining the
topology of Ea. If we put
(12.5.1) e«^(x,y) = da^(prxx9 ptay)
for each pair (x, y) of elements of E = ]^[ Ea, then it is immediately checked
ael
that the ea> A are pseudo-distances on E. The product of the topologies of the
Ea is then the topology on E defined by the ea^ as a runs through I and, for
each a e I, A runs through La. The set E endowed with this topology is called
the product of the spaces Ea. When I = {1,2} this definition agrees with that
given above for two spaces E1? E2.
For every finite family (a£, Ai)1^i^n and every finite family (r^igign °f
strictly positive real numbers, the set B(x; ((a£, A£)), (rf)) can be written in
the form f| Ba where, if a = af, we have Ba = B(praix; (A,-), (r,-)), where j
ael
runs through the indices for which a,- = a£, and Ba = Ea if a is not equal to
any oct. From this and (3.6.4) we deduce immediately:
(12.5.2) For each ael, the projection pra is a continuous mapping 0/E onto Ea,
and the image under pra of any open set in E is an open set in Ea.
(12.5.3) For each finite subset H of I, and each family (Ua)aeH, -where Ua is
open in Ea/0r each a e H, the set I fj Ua\ x / fj Ea\ is open in E.
\aeH / \aeI-H /
For it is the intersection of the sets pr^UJ, where a e H, and these sets
are open (12.5.2) and finite in number.
The sets described in (12.5.3) are called elementary sets in E. The descrip-
tion given above of the neighborhoods shows that (having regard to (3.9.3))
these sets form a basis for the topology of E. Incidentally, this shows also that
the topology of E depends only on the topologies of the Ea and not on the
families of pseudo-distances (rfa> A)- Moreover, if Sa is a basis for the topology
of Ea, the elementary sets such that Ua e 23a for all a form a basis for the
topology of E. It should be noted that, if I is infinite, a product f] Ua with
ael
Ua open in Ea, nonempty and ^ Ea for all ael, is not open in E.
(12.5.4) Let Aa be a subset of Ea, for each a, and let A = f] Aa. Then
ael
A = £| Aa. In particular, if each Aa is closed in Ea, then A is closed in E.
aelo-distances on E, and the definition of neigh-